Monotonic Variable Consistency Rough Set Approaches

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Monotonic Variable Consistency Rough Set Approaches

Jerzy Błaszczyn´ski

a

, Salvatore Greco

b

, Roman Słowin´ski

a,c,*

, Marcin Szel g

a a

Institute of Computing Science, Poznan´ University of Technology, 60-965 Poznan´, Poland b

Faculty of Economics, University of Catania, Corso Italia, 55, 95129 Catania, Italy c

Systems Research Institute, Polish Academy of Sciences, 01-447 Warsaw, Poland

a r t i c l e

i n f o

Article history: Received 11 July 2008

Received in revised form 9 January 2009 Accepted 25 February 2009

Available online 14 March 2009

Keywords: Rough sets

Dominance-based Rough Set Approach Monotonicity Bayesian conﬁrmation Rough membership Likelihood Accuracy of approximation Variable precision Variable consistency

a b s t r a c t

We consider probabilistic rough set approaches based on different versions of the defini-tion of rough approximadefini-tion of a set. In these versions, consistency measures are used to control assignment of objects to lower and upper approximations. Inspired by some basic properties of rough sets, we find it reasonable to require from these measures several prop-erties of monotonicity. We consider three types of monotonicity propprop-erties: monotonicity with respect to the set of attributes, monotonicity with respect to the set of objects, and monotonicity with respect to the dominance relation. We show that consistency measures used so far in the definition of rough approximation lack some of these monotonicity prop-erties. This observation led us to propose new measures within two kinds of rough set approaches: Variable Consistency Indiscernibility-based Rough Set Approaches (VC-IRSA) and Variable Consistency Dominance-based Rough Set Approaches (VC-DRSA). We investi-gate properties of these approaches and compare them to previously proposed Variable Precision Rough Set (VPRS) model, Rough Bayesian (RB) model, and previous versions of VC-DRSA.

1. Introduction

Calculation of rough set approximations can be the first step of the analysis of data. It allows to identify consistent data and put them into lower approximations of sets (concepts, classes, or unions of ordered classes). The following step is usually related to generalization of data. When we consider a classification problem, this step consists in induction of a classifier that can be further used for prediction. In the original rough set approach proposed by Pawlak[19,20], and in the dominance-based rough set approach proposed by Greco et al.[8,9,11,24], the lower approximation of a set is defined by a strict inclusion relation of some granules of knowledge in the approximated set. The lower approximation is thus composed of the granules that are subsets of the approximated set. Other granules are not included into lower approximation, regardless of the size of their overlap with the set and/or its complement. This definition of the lower approximation appears to be too restrictive in practical applications. In consequence, lower approximations of sets are often empty, preventing generalization of data in terms of relative certainty. This observation has motivated research on probabilistic generalizations of rough sets. Different versions of probabilistic rough set approaches were proposed, starting from Variable Precision Rough Set (VPRS) model[26,28,29], Variable Consistency Dominance-based Rough Set Approaches (VC-DRSA)[1,9,10], Bayesian Rough Set model and Rough Bayesian (RB) model[25,26], decision theoretic rough set model[13,30,31]and Parameterized Rough Sets

[14]. The probabilistic rough set approaches allow to extend lower approximation of a set by objects with sufﬁcient evidence

*Corresponding author. Address: Institute of Computing Science, Poznan´ University of Technology, 60-965 Poznan´, Poland.

E-mail addresses:[email protected](J. Błaszczyn´ski),[email protected](S. Greco),[email protected](R. Słowin´ski),mszelag@ cs.put.poznan.pl(M. Szel g).

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International Journal of Approximate Reasoning

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j a r

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for membership to the set. To quantify this evidence, the authors propose different measures of the overlap between a granule of knowledge based on a considered object and the approximated set or its complement. We call such measures consistency measures.

Inspired by some basic properties of rough sets, we ﬁnd it reasonable to require from consistency measures several prop-erties of monotonicity that correspond directly to monotonicity propprop-erties of the lower approximation. The present paper focuses on three types of monotonicity properties. These monotonicity properties are considered in various dimensions of the analyzed data set and are related to:

(1) extension of the set of attributes, (2) extension of the set of objects,

(3) extension of the union of ordered classes, (4) improvement of evaluation of an object.

Monotonicity in dimension (1) requires that precisiation of the description of objects by addition of attributes can only give more evidence for the assignment of these objects to the approximated set. Precisiation means here a more detailed descrip-tion of objects, without considering a semantic value of the addidescrip-tional informadescrip-tion. Let us observe that if a semantic value of additional attributes would be considered, then the precisiation could decrease the evidence for the assignment of objects to the approximated set. For example, a semantic value of additional attributes could depend on whether the precisiation by these attributes decreases or increases the confusion related to the assignment of objects to the approximated set. Then, in the first case, the semantic value would be considered positive, and in the second, negative. Thus, additional attributes with a negative semantic value would not increase the evidence for the assignment of objects to the approximated set. Monotonicity in dimension (1) is concordant with monotonicity of the accuracy of approximation defined by Pawlak[20]. This type of monotonicity is desirable for reduction of attributes in probabilistic rough set approaches. From classification perspective, monotonicity in this dimension corresponds to reasonable outcome of classifiers that are induced from data sets that overlap in the dimension of attributes (i.e., multiple classifiers generated on overlapping subsets of attributes from the extended set of attributes).

Monotonicity in dimension (2) requires that extension of the approximated set by addition of new objects, should not neg-atively affect the evidence for membership of the ‘‘old” objects to the approximated set. From classification perspective, this property allows to generate compatible classifiers on overlapping subsets of objects from the extended set of objects (i.e., incremental classifiers or ensembles of classifiers created on overlapping sets of objects, like in bagging or boosting). Let us observe that monotonicity in dimension (2) may be discussed in the context of Bayesian confirmation theory, i.e., the theory which studies how a piece of evidence E provides ‘‘evidence for or against” or ‘‘support for or against” hypothesis H (for an extensive survey see[5]). In fact, we can imagine that new objects constitute new evidence which may confirm or disconfirm the hypothesis that an object can be assigned to an approximated set. Let us explain this point in terms of the famous paradox, called black raven paradox[15]. Consider hypothesis H ‘‘all ravens are black”, which corresponds to the idea of assigning to set ‘‘black” all objects having value ‘‘raven” on attribute ‘‘raven yes or not”. Hypothesis H can be read as the implication ‘‘if an object is a raven, than it is black”. Within the rough set approach (for a discussion about relationships between Bayesian con-firmation theory and rough set theory see[12]), the hypothesis concerns the membership to decision class ‘‘black” of those objects which according to the condition attribute are ‘‘raven”. At a first look, one can imagine that any object being ‘‘raven” and ‘‘black” confirms the hypothesis, any object being ‘‘raven” and ‘‘non-black” disconfirms the hypothesis, and all objects being not ‘‘raven” do not confirm and do not disconfirm the hypothesis. In general, this observation can be expressed as fol-lows: an object confirms an implication if and only if it satisfies both the premise and the conclusion of the implication (i.e., it is ‘‘black” and ‘‘raven”); it disconfirms the implication if and only if it satisfies the premise, but not the conclusion (i.e., it is ‘‘non-black” but it is ‘‘raven”); it does not confirm and does not disconfirm the implication if it does not satisfy the premise (i.e., it is not ‘‘raven”). In this perspective, each new black object cannot disconfirm the hypothesis. In fact, it can confirm the hypothesis if it is also ‘‘raven” or neither confirm nor disconfirm if it is not ‘‘raven”. In our context, this means that extending the approximated set of black objects, we cannot reduce the membership of an object to the considered set, and this agrees with monotonicity in dimension (2). Hempel observed in[15]that hypothesis H is logically equivalent to the implication ‘‘if an object is non-black, then it is not raven”, which is confirmed by objects being ‘‘non-black” and not ‘‘raven”. Remark that this observation leads to the paradox that pink socks can confirm the hypothesis that ‘‘all ravens are black”. Also in this case, a black object cannot disconfirm the hypothesis (even if it cannot also confirm it), because it does not satisfy the premise. In our context, this agrees again with monotonicity in dimension (2). Observe, however, that in case of probabilistic confirma-tion, some authors find it reasonable to expect that ‘‘black non-ravens” can reduce the confirmation degree[12]. In this case, a considered confirmation measure of the hypothesis ‘‘ifU, thenW”, can be expressed as credibility of the proposition ‘‘ifWis satisfied more frequently whenUis satisfied rather than whenUis not satisfied”. According to this understanding, black ravens and non-black non-ravens confirm the hypothesis, while non-black ravens and black non-ravens disconfirm the hypothesis. With respect to black non-ravens, they disconfirm the hypothesis because they increase the probability thatW

is satisfied whenUis not satisfied, i.e., they increase the probability that an object is black when it is not raven. In this sense, expectations for probabilistic confirmation do not agree with monotonicity in dimension (2).

Finally, monotonicity with respect to the dominance relation is considered in dimensions (3) and (4). Monotonicity in these dimensions concerns data sets with speciﬁed orders of preference. They allow to generate classiﬁers that permit to

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make classiﬁcation decisions respecting preference orders. Property (m3) is related to an important property of DRSA, which wants that a lower approximation of any upward (downward) union of ordered classes includes a lower approximation of any of its upward (downward) sub-unions. Property (m4) ensures that if an object belongs to a lower approximation of an upward (downward) union of ordered classes, then all objects from this union which dominate (are dominated by) this object will also belong to the lower approximation.

At the end of this introduction, it is worth noting, however, that instead of requiring monotonicity properties from con-sistency measures, one could accept monotonic behavior of concon-sistency measures and consider application of non-monotonic logic[3,7,18]. This way of looking at probabilistic generalizations of rough sets could be an interesting subject for future research.

In the next section, we remind basic definitions of original Indiscernibility-based Rough Set Approach and Dominance-based Rough Set Approach. Then, we define monotonicity properties required for consistency measures that are used in Monotonic Variable Consistency Rough Set Approaches. In Section3, we show which of the monotonicity properties are sat-isfied by consistency measures that were used in probabilistic rough set approaches proposed so far. We also give examples of shortcomings of these measures. In Section 4, we define new monotonic Variable Consistency Indiscernibility-based Rough Set Approaches (VC-IRSA). Along the way, two types of monotonic consistency measures are introduced. We prove and interpret their properties. In Section5, we show how the measures defined for the indiscernibility relation can be refor-mulated for the dominance relation. In consequence, new monotonic Variable Consistency Dominance-based Rough Set Approaches (VC-DRSA) are proposed. Finally, we present an illustrative example for indiscernibility-based and domi-nance-based approaches. We conclude by giving remarks and recommendations for applications of the new approaches. 2. Monotonicity properties required for rough set approaches

In the rough set approach, classiﬁcation of object y from universe U to a given set X # U is based on available data. Data is presented as a decision table, where rows correspond to objects from U and columns correspond to attributes from a ﬁnite set A. Among attributes from set A there are attributes with preference-ordered value sets, called criteria, and regular attri-butes whose value sets are not preference-ordered. Moreover, the set of attriattri-butes A is divided into disjoint sets of condition attributes C and decision attributes D. For simplicity, we assume set D to be a singleton D ¼ fdg.

The decision attribute d makes a partition of set U into a finite number of disjoint sets of objects, called decision classes. Let X # U be one of these decision classes. Decision about classification of object y 2 U to set X depends on its class label known from the decision table, and/or on its relation with other objects from the table. In the original rough set approach, the considered relation is the indiscernibility relation[19,20]. For this reason, we call this approach Indiscernibility-based Rough Set Approach (IRSA). Consideration of the indiscernibility relation is meaningful when set of attributes A is composed of regular attributes only. Indiscernibility relation makes a partition of universe U into disjoint blocks of objects that have the same description and are considered indiscernible. Such blocks are called granules. Let Vaibe the value set of attribute ai2 C and f : U C ! Vaibe a total function such that f ðx; aiÞ 2 Vai. Indiscernibility relation IPis defined for a non-empty subset of attributes P # C as

IP¼ fðy; zÞ 2 U U : f ðy; aiÞ ¼ f ðz; aiÞ for all ai2 Pg:

Moreover, IPðyÞ denotes a set of objects indiscernible with object y using set of attributes P. It is called a granule of

P-indis-cernible objects.

When condition attributes from C and decision attribute d have preference-ordered value sets, in order to make mean-ingful classiﬁcation decisions, one has to consider the dominance relation instead of the indiscernibility relation. It has been proposed in[8,9,11,24]and the resulting approach was called Dominance-based Rough Set Approach (DRSA). Dominance relation makes a partition of universe U into granules being dominance cones. The dominance relation DP is deﬁned for a

non-empty subset of criteria P # C as

DP¼ fðy; zÞ 2 U U : f ðy; aiÞ f ðz; aiÞ for all ai2 Pg;

where f ðy; aiÞ f ðz; aiÞ means ‘‘y is at least as good as z with respect to (w.r.t.) criterion ai”. Dominance relation DPis a partial

preorder (i.e. reﬂexive and transitive). For each object y 2 U two dominance cones (granules) are deﬁned w.r.t. P # C. The P-positive dominance cone Dþ

PðyÞ is composed of all objects that are dominating y. The P-negative dominance cone D PðyÞ

is composed of all objects that are dominated by y. Formal deﬁnitions of dominance cones are as follows:

Dþ

pðyÞ ¼ fz 2 U : zDPyg;

D

pðyÞ ¼ fz 2 U : yDPzg:

We are considering a classiﬁcation problem with n disjoint classes. While in IRSA, decision classes Xi, i ¼ 1; . . . ; n, are not

necessarily ordered, in DRSA, they are ordered, such that if i < j, then class Xiis considered to be worse than Xj. Moreover,

DRSA takes into account monotonic relationships between evaluations of objects on particular criteria and assignment of these objects into decision classes. For example, the better the value of criterion ai2 C for object y, the better the decision

class it may belong. From this follows the dominance principle which says that if evaluations of object y on all considered criteria are not worse than evaluations of object z, then y should be assigned to a class not worse than z. Violation of this

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principle causes inconsistency in the data table which is captured within DRSA by approximations of sets. In order to handle preference orders, and monotonic relationships between evaluations on criteria and assignment to decision classes, approx-imations made in DRSA concern the following unions of decision classes: upward unions XP

i ¼

S

tPiXt, where i ¼ 2; 3; . . . ; n,

and downward unions X6 i ¼

S

t6iXt, where i ¼ 1; 2; . . . ; n 1.

One of the most important features of rough set approaches is the separation of knowledge which is consistent, from knowledge which is possibly inconsistent. In IRSA, a key point is to ﬁnd evidence for assignment of objects to particular deci-sion classes Xi. In DRSA, the key point is to ﬁnd evidence for assignment of objects to unions of decision classes XPi and X

6 i.

In order to avoid repetition of the same deﬁnitions and properties for IRSA and DRSA, we will use a unique symbol X to denote a set of all objects belonging to class Xi, in the context of IRSA, or to union of classes XPi , X

6

i, in the context of DRSA.

Let us specify conditions that must be satisﬁed by consistency measures. We distinguish gain-type and cost-type consis-tency measures. First, let us consider y1;y22 U, P # C, X # U. Given description of y1and y2by P:

a gain-type consistency measure fP

XðyÞ is any measure satisfying condition: fXPðy1Þ P fXPðy2Þ () it is not less likely that y1

belongs to X, than that y2belongs to X,

a cost-type consistency measure gP

XðyÞ is any measure satisfying condition: gPXðy1Þ 6 gPXðy2Þ () it is not less likely that y1

belongs to X, than that y₂belongs to X.

Second, let us consider y 2 U, P # C, X; Y # U, where Y has the same interpretation as X (i.e., it denotes a class or a union of classes). Given description of y by P:

a gain-type consistency measure fP

XðyÞ is any measure satisfying condition: fXPðyÞ P fYPðyÞ () it is not less likely that y

belongs to X, than that it belongs to Y. a cost-type consistency measure gP

XðyÞ is any measure satisfying condition: gPXðyÞ 6 gPYðyÞ () it is not less likely that y

belongs to X, than that it belongs to Y.

A consistency measure expresses the evidence for membership to set X. For a gain-type measure, the higher the value, the more consistent is the given object. For a cost-type measure, the lower the value, the more consistent is the given object. In this paper, we investigate desirable properties of consistency measures.

Each set X, may include objects for which, due to inconsistency, we are unable to ﬁnd enough evidence for their membership to X. In such a case, we can approximate set X by two sets, the P-lower approximation and the P-upper approximation of X, where P # C. Let us give generic deﬁnitions of P-lower approximations of set X, which involve consistency measures fP

XðyÞ or gPXðyÞ.

For P # C; X # U; y 2 U, given a gain-type consistency measure fP

XðyÞ and a gain-threshold

a

X, we get the following

deﬁ-nitions of P-lower approximation of set X:

PaX_{ðXÞ ¼ fy 2 U : f}P

XðyÞ P

a

Xg ð1Þ

or PaX_{ðXÞ ¼ fy 2 X : f}P

XðyÞ P

a

Xg: ð2Þ

Analogically, given a cost-type consistency measure gP

XðyÞ and a cost-threshold bX, we get the following deﬁnitions: PbX_{ðXÞ ¼ fy 2 U : g}P

XðyÞ 6 bXg ð3Þ

or PbX_{ðXÞ ¼ fy 2 X : g}P

XðyÞ 6 bXg: ð4Þ

In the above deﬁnitions, gain-threshold

a

X2 ½0; AX and cost-threshold bX2 ½0; BX. These thresholds are parameters

depend-ing on the interpretation of the gain-type or cost-type consistency measure, respectively. They play the role of technical parameters inﬂuencing the degree of consistency of objects belonging to lower approximation of X.

Thus, the values of AXand BXalso depend on the interpretation of the corresponding consistency measure. For example, in

case of probabilistic P-lower approximation deﬁned using the rough membership measure, AX¼ 1 and value of

gain-thresh-old

a

X2 ½0; 1 can be calculated using method presented in[13,30]. This method is based on application of the Bayesian

deci-sion procedure in transformation of risk into the value of

a

X.

The above deﬁnitions of P-lower approximations relax the non-parametric deﬁnitions. Precisely, the non-parametric def-inition for IRSA and class Xiis as follows:

PðXiÞ ¼ fy 2 U : IPðyÞ # Xig ¼ fy 2 Xi:IPðyÞ # Xig;

and for DRSA, and unions of classes XP i , X 6 i, it is as follows: PðXP i Þ ¼ fy 2 U : D þ PðyÞ # XPi g ¼ fy 2 XPi :D þ PðyÞ # XPi g; PðX6 iÞ ¼ fy 2 U : D PðyÞ # X 6 i g ¼ fy 2 X 6 i :D PðyÞ # X 6 ig:

An obvious condition of this relaxation is:

PðXÞ # PaX_ðXÞ; _ð5Þ

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The deﬁnition of P-upper approximation and the deﬁnition of P-boundary of set X make use of the complementarity property of rough approximations, and are the same for all the approaches considered in this work.

For P # C; X; :X # U, where :X ¼ U X, P-upper approximation of set X is deﬁned as

PaX_{ðXÞ ¼ U P}aX_ð:XÞ; _PbX_{ðXÞ ¼ U P}bX_ð:XÞ; _ð7Þ

while P-boundary of set X is deﬁned as

BnaX

P ðXÞ ¼ P

aX_{ðXÞ P}aX_ðXÞ; _BnbX

P ðXÞ ¼ P

bX_{ðXÞ P}bX_ðXÞ: _ð8Þ

Let us remark that the notion of consistency was also used in IRSA, to measure consistency of the whole decision table

[4,16,22,23]. In this case, different instances of the entropy measure were applied instead of the quality of approximation.

Entropy measures were also applied to deﬁne consistency of a granule composed of P-indiscernible objects[23]. In the case of the whole decision table, as well as in the case of a single granule, consistency was considered with respect to all possible classes from the decision table.

In the present paper, we understand consistency in a different way. We consider consistency of particular objects with respect to the approximated sets.

One can observe that properties of rough approximations deﬁned above depend on properties of consistency measures fP

XðyÞ and gPXðyÞ. Thus, it is possible to formulate some properties with respect to these measures, which ensure desirable

properties of rough approximations.

For IRSA and DRSA, it is reasonable to require that consistency measures fP

XðyÞ and gPXðyÞ fulﬁll the following properties of

monotonicity (henceforth called monotonicity properties):

(m1) Monotonicity w.r.t. set of attributes P # C. Formally, for all P # P0_#_{C, X # U, y 2 U, a gain-type measure f}P

XðyÞ is

mono-tonically non-decreasing w.r.t. P, if and only if (iff)

fP XðyÞ 6 fP

0

XðyÞ; ð9Þ

and a cost-type measure gP

XðyÞ is monotonically non-increasing w.r.t. P, iff gP

XðyÞ P g P0

XðyÞ: ð10Þ

(m2) Monotonicity w.r.t. set of objects X # U, when set X is augmented by new objects. Formally, for all P # C, X # U, X0

¼ X [ XD, XD

\ U ¼ ;, y 2 U, a gain-type measure fP

XðyÞ is monotonically non-decreasing w.r.t. X, iff fP

XðyÞ 6 f P

X0ðyÞ; ð11Þ

and a cost-type measure gP

XðyÞ is monotonically non-increasing w.r.t. X, iff gP

XðyÞ P g P

X0ðyÞ: ð12Þ

Moreover, for DRSA, it is reasonable to require that measures fP XP i ðyÞ (or fP X6 i ðyÞ) and gP XP i ðyÞ (or gP X6 i

ðyÞ) fulﬁll the following monotonicity properties:

(m3) Monotonicity w.r.t. union of classes XP

i #U and X 6

k#U. Formally, for all P # C, X P i #X P j #U, j 6 i, X 6 k#X 6 l #U, l P k, y 2 U, gain-type measures fP XP i ðyÞ and fP X6 k

ðyÞ are monotonically non-decreasing w.r.t. XP i and X 6 k, respectively, iff fP XP i ðyÞ 6 f P XP j ðyÞ; f P X6 kðyÞ 6 f P X6 lðyÞ: ð13Þ

Analogously, a cost-type measures gP XP

i

ðyÞ and gP X6

k

ðyÞ are monotonically non-increasing w.r.t. XP i and X 6 k, respectively, iff gP XP i ðyÞ P g P XP j ðyÞ; g P X6 kðyÞ P g P X6 lðyÞ: ð14Þ

(m4) Monotonicity w.r.t. P-dominance relation, P # C. Formally, for all P # C, XP i ;X

6

i #U, y 2 U, and * standing for either P

or 6 in every instance, a gain-type measure fP X

iðyÞ is monotonically non-decreasing w.r.t. P-dominance relation, iff

8

y1;y22 U : y1DPy2) fXP

iðy1Þ P f

P X

iðy2Þ; ð15Þ

and a cost-type measure gP X

iðyÞ is monotonically non-increasing w.r.t. P-dominance relation, iff

8

y1;y22 U : y1DPy2) gPX

iðy1Þ 6 g

P X

iðy2Þ: ð16Þ

Monotonicity properties (m1) and (m2) are related to the basic properties of rough sets. Monotonicity properties (m3) and (m4) are specific to DRSA. A rough set approach is called monotonic when the consistency measure used to define its lower approximation fulfills relevant monotonicity properties. For IRSA, relevant properties are (m1) and (m2), while for DRSA, rel-evant properties are (m1), (m2), (m3) and (m4).

Property (m1) is particularly important. Property (m1) of measures fP

XðyÞ and gPXðyÞ ensures monotonicity of P-lower

approximation w.r.t. set of attributes P # C, deﬁned according to(2) and (4), respectively. This property imposes that addi-tional information about objects from U can only give more evidence for the observed assignment of objects to classes. In this

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case, additional information means a precisiation by more detailed description of considered objects using an extended set of attributes. Property (m1) is also concordant with the observation that additional attributes can only decrease comparability in the set of objects. When less objects are comparable, then also less inconsistent assignments to classes is observed.

Property (m2) of measures fP

XðyÞ and gPXðyÞ ensures monotonicity of P-lower approximation w.r.t. set of objects X # U.

Property (m2) states that when we consider two sets of objects X0

X, the evidence for membership to X0for objects from X should not be worse than the evidence for their membership to X. In other words, extension of class Xior union of classes

XP i (X

6

i) by addition of new objects, should not negatively affect the evidence for membership of the objects to the extended

class or union of classes.

In DRSA, property (m3) of measures fP XP i ðyÞ (or fP X6 i ðyÞ) and gP XP i ðyÞ (or gP X6 i

ðyÞ) ensures monotonicity of P-lower approxima-tion w.r.t. union XP

i #U (or X 6

i #U). This property states that value of a gain-type consistency measure for a union that is a

superset should not decrease, while value of a cost-type consistency measure should not increase. For example, for object y which belongs to upward unions XP

i and X P j , where X P i #X P

j #U, value of gain-type consistency measure fXPP j

ðyÞ should not be worse than the value of this measure calculated for union XP

i .

The importance of property (m4) in Variable Consistency DRSA (VC-DRSA) was already discussed in[1], however, under the name of monotonicity of membership to lower approximation. Monotonicity w.r.t. P-dominance relation, P # C, is a very desirable property for a measure used in the deﬁnition of P-lower approximation of union X

i, where * stands for either P

or 6. In case of deﬁnitions based on formula(2), where it is checked if fP X

iðyÞ P

a

X

i, a consistency measure deﬁned for X

P i

should satisfy(15), while a consistency measure deﬁned for X6

i should satisfy(16). For deﬁnitions based on formula(4),

where it is checked if gP X

iðyÞ 6 bX

i, a consistency measure deﬁned for X

P

i should satisfy(16), while a consistency measure

deﬁned for X6

i should satisfy(15). This ensures a kind of continuity of lower approximations – as soon as some object

y 2 XP

i is included in the P-lower approximation of union X P

i , every object z 2 X P

i , which P-dominates y, will also be included

in this approximation. Analogically, if some object y 2 X6

i is included in P-lower approximation of union X 6

i, then every

object z 2 X6

i, which is P-dominated by y, will also belong to the considered approximation.

3. Are rough membership, conﬁrmation measures and Bayes factor monotonic consistency measures?

Rough membership measure was introduced in[27]and its properties were further investigated in[21,31]. It is used to control positive regions in Variable Precision Rough Set (VPRS) model[26,28,29]and in previous versions of Variable Con-sistency Dominance-based Rough Set Approaches (VC-DRSA)[1,9,10]. Rough membership was also considered in the context of attribute reduction[17].

In IRSA, rough membership of y 2 U to X # U w.r.t. P # C is deﬁned as

l

P XðyÞ ¼ IPðyÞ \ X j j IPðyÞ j j :

Rough membership is a gain-type consistency measure. It captures a ratio of objects that belong to granule IPðyÞ and to

con-sidered set X, among all objects belonging to granule IPðyÞ. For example, if we would consider a medical diagnosis, the value

of rough membership would express the ratio of the number of patients that have the same symptoms and suffer from the considered disease to the number of all patients that have the same symptoms. This measure can also be treated as an esti-mate of conditional probability Prðx 2 Xjx 2 IPðyÞÞ. In IRSA, rough membership is used in deﬁnition(1), and it is expected to

have properties (m1) and (m2). Unfortunately, property (m1) does not hold, which is shown by the example presented in

Fig. 1. First, we consider attribute a1only. All objects have the same value on that attribute (i.e., they all belong to the same

granule). Thus,

l

fa1g

X2 ðy1Þ ¼

l

fa1g

X2 ðy2Þ ¼

l

fa1g

X2 ðy3Þ ¼ 0:66. Second, we consider set P ¼ fa1;a2g. Then, we have two granules. The ﬁrst one consists of objects y1;y2and the other one is composed of object y3. The value of rough membership to class X2

drops to 0:5 in the ﬁrst granule. On the other hand, property (m2) holds for rough membership measure

l

P

XðyÞ (seeProof

of Theorem 4.8in the Appendix).

Other measures than rough membership have also been used in rough set approaches. For example, conﬁrmation mea-sures[5,12]were considered together with rough membership in Parameterized Rough Sets (PRS)[14]. Conﬁrmation mea-sures quantify the degree to which membership of object y to given granule IPðyÞ provides ‘‘evidence for or against” or

‘‘support for or against” assignment to considered set X. They are gain-type consistency measures and according to[14], they are used within definition(1). Confirmation measures should have properties (m1) and (m2). Unfortunately, as it may be shown, the well-known confirmation measures do not have property (m1).

The Bayes factor has similar properties to conﬁrmation measures (its formulation is close to the conﬁrmation measure l

[5]). It is a gain-type consistency measure used in the Rough Bayesian (RB) model[25]. The Bayes factor for y 2 U and X # U, w.r.t. P # C, is deﬁned as BP XðyÞ ¼ jIPðyÞ \ Xjj:Xj jIPðyÞ \ :XjjXj :

The Bayes factor is a ratio of estimates of two conditional probabilities Prðx 2 IPðyÞjx 2 XÞ and Prðx 2 IPðyÞjx 2 :XÞ. Coming

back to the example with medical diagnosis, the Bayes factor would express, in this case, the ratio of the estimate of prob-ability that a patient has the considered symptoms on condition that he suffers from the considered disease to the estimate

(7)

of probability that he has these symptoms on condition that he does not suffer from this disease. This measure is used in deﬁnition(1)and it is expected to have properties (m1) and (m2). Unfortunately, this is not the case. Let us come back to the example presented inFig. 1. First, let us observe that Bfa1g

X2 ðy2Þ ¼ 1, while B

P X2ðy2Þ ¼

1

2. This shows that the Bayes factor does

not have property (m1). Second, let us extend the set of objects with one new object y4, which belongs to class X2and has the

following description: a1¼ 0, a2¼ 1. We can notice that BPX2ðy2Þ ¼

1 2and B P X0 2ðy2Þ ¼ 1 3, where X 0

2¼ fy2;y3;y4g. This shows that

the Bayes factor also does not have property (m2).

Now, let us consider DRSA. In this case, rough membership is deﬁned for P # C, XP

;X6 #U, y 2 U, as

l

P XPðyÞ ¼ Dþ PðyÞ \ X P Dþ PðyÞ ;

l

PX6ðyÞ ¼ D PðyÞ \ X 6 D PðyÞ ; where XP , X6

denote upward and downward unions of decision classes, respectively. Values of rough membership

l

P XPðyÞ and

l

P

X6ðyÞ can be interpreted as estimates of probability Prðz 2 X

P

jzDPyÞ and Prðz 2 X 6

jyDPzÞ, respectively.

Formulation of the Bayes factor for P # C, XP

;X6 #U, y 2 U, is as follows: BP XPðyÞ ¼ jDþ PðyÞ \ X P_jj:XP_j jDþ PðyÞ \ :X P_jjXP_j; B P X6ðyÞ ¼ jD PðyÞ \ X 6 jj:X6 j jDPðyÞ \ :X 6 jjX6 j:

Both these measures are gain-type and they are used within DRSA in deﬁnition(2). They are expected to have properties (m1), (m2), (m3) and (m4). Measure

l

P

XPðyÞ (or

l

P_X6ðyÞ) has property (m2) – seeProof of Theorem 5.20(or5.21) in the Appen-dix. It also can be shown that measure

l

P

XPðyÞ (or

l

P_X6ðyÞ) has property (m3). Unfortunately, measure

l

P

XPðyÞ (or

l

P_X6ðyÞ) has neither property (m1) nor (m4). Moreover, measure BP

XPðyÞ (or BP_X6ðyÞ) has none of the monotonicity properties considered in this paper. Let us illustrate the lack of monotonicity by the example shown inFig. 2. First, let us consider measure

l

P

XPðyÞ. We can notice that

l

fa2g

XP 3 ðy2Þ ¼ 3 4, while

l

P XP 3ðy2Þ ¼ 2 3. Since

l

fa2g XP 3 ðy2Þ >

l

P XP 3ðy2Þ, measure

l

P

XPðyÞ does not have property (m1). More-over,

l

P XP 3ðy1Þ ¼ 1 2and

l

P XP 3ðy2Þ ¼ 2

3, which shows that measure

l

P

XPðyÞ also does not have property (m4). Second, let us con-sider measure BPXPðyÞ. We can notice that Bfa2g

XP 3 ðy2Þ ¼ 2, while B P XP 3ðy2Þ ¼ 4 3. Since B fa2g XP 3 ðy2Þ > B P XP 3ðy2Þ, measure B P XPðyÞ does not have property (m1). In order to show that measure BPXPðyÞ does not have property (m2), let us assume that object y₃ is not originally present in the considered data set and is added as a new object. We can observe that BP

XP 3ðy2Þ ¼ 2 > BP X0P 3ðy2Þ ¼ 4 3, for X P

3 ¼ fy1;y2g and X0P3 ¼ fy1;y2;y3g. Now, let us calculate Bayes factors for object y2and unions of classes

XP 2, X P 3. We have B P XP 3ðy2Þ ¼ 4 3>B P XP 2ðy2Þ ¼ 1

2. This shows that measure B P

XPðyÞ does not have property (m3). Finally, let us notice that BP XP 3ðy2Þ ¼ 4 3>B P XP 3ðy1Þ ¼ 2

3. This proves, that measure B P

XPðyÞ also does not have property (m4). 4. Monotonic Variable Consistency Indiscernibility-based Rough Set Approaches

Our motivation for proposing Variable Consistency Indiscernibility-based Rough Set Approaches (VC-IRSA) comes from the need of ensuring monotonicity of lower approximations w.r.t. set of attributes. Due to the deﬁnition of the upper approx-imation based on complementarity, w.r.t. the lower approxapprox-imation, this monotonicity property also concerns the upper approximation. The main difference between VC-IRSA and VPRS[26,28,29], RB model[25]and PRS[14]is that in VC-IRSA one considers for inclusion to P-lower approximations only these objects which belong to the approximated set (deﬁnitions

(2) and (4)). In VPRS, RB model and PRS whole granules are included to P-lower approximations (deﬁnitions(1) and (3)).

Remark that a granule included in a P-lower approximation may be composed of some inconsistent objects. After enlarging a1 a2 ● ● ● ● ● ●

●

0 1 0 1 y1 y2 y3 X1 X2 X2

(8)

set P of attributes to P0_{P, some of P-indiscernible and inconsistent objects may become P}0_{-discernible and thus consistent,}

so, if we would like to preserve monotonicity of lower approximations, then we should keep in the P0_{-lower approximation}

the P0_{-discernible objects that do not belong to the approximated set. This, is not reasonable, however. Motivated by this}

remark, we consider only lower approximations deﬁned according to(2)or(4).

Below, we introduce new consistency measures for VC-IRSA. We also present theorems concerning monotonicity prop-erties of these measures. Proofs of all the theorems are given in the Appendix.

As it was already mentioned in Section2, monotonicity properties of a consistency measure used in the deﬁnition of the P-lower approximation imply monotonicity properties of this approximation.

4.1. Consistency measure

The ﬁrst consistency measure that we consider in VC-IRSA is a cost-type measure

P

XiðyÞ. For P # C; Xi;:Xi#U, where :Xi¼ U Xi, y 2 U, it is deﬁned as

P XiðyÞ ¼ jIPðyÞ \ :Xij j:Xij : ð17Þ

In the numerator of(17)there is the number of objects in U that do not belong to class Xiand are indiscernible with object y.

In the denominator of(17)there is the number of objects in U that do not belong to class Xi. The ratio

PXiðyÞ is an estimate of conditional probability Prðx 2 IPðyÞjx 2 :XiÞ, called also a catch-all likelihood[6]. This measure is an estimate of probability

that object y belongs to granule IPðyÞ given that it does not belong to class Xi. It may result in low values of consistency

mea-sure

P

XiðyÞ for classes Xithat have low cardinality. Theorem 4.1. Measure

P

XiðyÞ has property (m1), i.e., for all P # P

0_#_{C; X} i#U; y 2 U:

P XiðyÞ P

P0 XiðyÞ: Theorem 4.2. Measure

P

XiðyÞ has property (m2). More precisely, for all P # C, Xi#U, X

0 i¼ Xi[ XDi, X D i \ U ¼ ;, y 2 U:

P XiðyÞ ¼

P X0 iðyÞ:

Monotonic P-lower approximation of class Xideﬁned according to(4)takes the form: Pb_Xi

ðXiÞ ¼ fy 2 Xi:

PXiðyÞ 6 bXig; ð18Þ

where cost-threshold bXi2 ½0; 1 reﬂects the highest degree of consistency acceptable to include object y in the P-lower approximation of class Xi.

Theorem 4.3. Lower approximation deﬁned according to(18)satisﬁes condition(6):

PðXiÞ # PbXiðXiÞ: 0 1 2 3 4 5 012345 a1 a2 ● ● ● ● ● ● ● ● ● ●

●

y1 y2 y3 y4 y5 X3 X3 X3 X2 X1

(9)

0

Another consistency measure that we consider in VC-IRSA is a cost-type measure

0P

XiðyÞ. For P # C; Xi;:Xi#U, where :Xi¼ U Xi, y 2 U, it is deﬁned as

0P XiðyÞ ¼ jIPðyÞ \ :Xij jXij : ð19Þ

In the numerator of(19)there is the number of objects in U that do not belong to class Xiand are indiscernible with object y.

In the denominator of(19)there is the number of objects in U that belong to class Xi. This measure represents the ratio of

objects z 2 U that are counterexamples to the implication z 2 IPðyÞ implies z 2 Xito the total number of objects in Xi. It lacks

the likelihood interpretation that we give for

P

XiðyÞ. It should be noticed that

0P

XiðyÞ may have low values for classes Xithat have high cardinality.

Theorem 4.4. Measure

0P

0 #C; Xi#U; y 2 U:

0P XiðyÞ P

0P0 XiðyÞ: Theorem 4.5. Measure

0P

XiðyÞ has property (m2). More precisely, for all P # C, Xi#U, X

0 i¼ Xi[ XDi, X D i \ U ¼ ;, y 2 U:

0P XiðyÞ ¼

0P X0 iðyÞ:

Monotonic P-lower approximation of class Xideﬁned according to(4)takes the form: Pb0Xi_ðX_i_{Þ ¼ fy 2 X}_i:

0P XiðyÞ 6 b 0 Xig; ð20Þ where cost-threshold b0 Xi2 0; j:Xij jXij h i

reﬂects the highest degree of consistency acceptable to include object y in the P-lower approximation of class Xi.

PðXiÞ # P b0

Xi_ðX_i_Þ:

l

A gain-type consistency measure that can be considered in VC-IRSA is measure

l

P

XiðyÞ. For P # C, Xi#U, y 2 U, it is deﬁned as

l

P XiðyÞ ¼ max_{R # P} IRðyÞ \ Xi j j IRðyÞ j j : ð21Þ Consistency measure

l

P

XiðyÞ is calculated as a maximum rough membership to class Xiover all subsets R of the set of attri-butes P.

Theorem 4.7. Measure

l

P

0_#_{C; X} i#U; y 2 U:

l

P XiðyÞ 6

l

P0 XiðyÞ: Theorem 4.8. Measure

l

P

XiðyÞ has property (m2), i.e., for all P # C, Xi#U, X

0 i¼ Xi[ XDi, X D i \ U ¼ ;, y 2 U:

l

P XiðyÞ 6

l

P X0 iðyÞ:

Monotonic P-lower approximation of class Xideﬁned according to(2)takes the form: Pa_Xi

ðXiÞ ¼ fy 2 Xi:

l

PXiðyÞ P

a

Xig; ð22Þ

where gain-threshold

a

Xi2 ½0; 1 reﬂects the lowest degree of consistency acceptable to include object y in the P-lower approximation of class Xi.

PðXiÞ # P

a_Xi

ðXiÞ:

In[2], we also considered a gain-type consistency measure

l

P

XiðyÞ which is deﬁned analogously to

l

P

XiðyÞ. For P # C, Xi#U, y 2 U:

(10)

l

P XiðyÞ ¼ min_RP IRðyÞ \ Xi j j IRðyÞ j j :

It appears that this measure also has properties (m1) and (m2). However, it was used to deﬁne the P-lower approximation together with

l

P

XiðyÞ. We refrain from using

l

P

XiðyÞ alone in the deﬁnition of the P-lower approximation. 4.4. Summary

In this section, we proposed deﬁnitions of three measures that ensure monotonicity of VC-IRSA. In Section4.1consistency measure

was introduced. This measure has the meaning of a likelihood that an object is not a member of the considered class, given that it belongs to a granule of indiscernible objects. Such a kind of likelihood is sometimes called a catch-all like-lihood. In Section4.2consistency measure

0 _{was introduced. This measure can be seen as complementary to measure}

_.

They differ only by denominator. Monotonic measure deﬁned in Section4.3involves rough membership measure

l

. It re-quires calculation of

l

over all subsets of P # C. For all of these measures, we checked monotonicity properties (m1) and (m2). The results are summarized inTable 1.

5. Monotonic Variable Consistency Dominance-based Rough Set Approaches

We reformulate deﬁnitions of monotonic approaches presented in Section4, replacing indiscernibility relation by dom-inance relation. Precisely, instead of granule IPðyÞ, we use positive dominance cone DþPðyÞ or negative dominance cone DPðyÞ,

and instead of decision class Xi, we consider upward union of decision classes XPi or downward union of decision classes X 6 i.

We also present theorems concerning monotonicity properties of the introduced consistency measures. Proofs of all the the-orems are given in the Appendix.

As it was already mentioned in Section2, monotonicity properties of a consistency measure used in the deﬁnition of the P-lower approximation imply monotonicity properties of this approximation.

Cost-type consistency measures

P XP i ðyÞ and

P X6 i ðyÞ, for P # C, XP i , X 6 i, X 6 i1, X P

iþ1#U, y 2 U, are deﬁned as

P XP i ðyÞ ¼ jDþPðyÞ \ X 6 i1j jX6 i1j ;

P X6 iðyÞ ¼ jDPðyÞ \ X P iþ1j jXP iþ1j : ð23Þ Consistency measure

P XP i ðyÞ (or

P X6 i

ðyÞ) can be interpreted as an estimate of conditional probability that object y belongs to the considered dominance cone given that it does not belong to the considered union. In other words, it is the number of objects in the dominance cone of object y that do not belong to the considered union of classes, divided by the number of all those objects that do not belong to the considered union of classes. Analogously to Section4.1, measures

P

XP i ðyÞ and

P X6 i

ðyÞ can be interpreted as catch-all likelihoods. Theorem 5.1. Measures

P

XP

i ðyÞ and

P X6

iðyÞ have property (m1), i.e., for all P # P

0_#_{C; X}P i ;X 6 i #U; y 2 U:

P XP i ðyÞ P

P0 XP i ðyÞ;

P X6 iðyÞ P

P0 X6 iðyÞ: Theorem 5.2. Measure

P XP i

ðyÞ has property (m2). More precisely, for all P # C, XP

i #U, X0Pi ¼ X P i [ X DP i , X DP i \ U ¼ ;, y 2 U:

P XP i ðyÞ ¼

P X0P i ðyÞ: Theorem 5.3. Measure

P X6

iðyÞ has property (m2). More precisely, for all P # C, X

6 i #U, X 06 i ¼ X 6 i [ X D6 i , X D6 i \ U ¼ ;, y 2 U:

P X6 iðyÞ ¼

P X06 i ðyÞ: Table 1

Monotonicity of consistency measures deﬁned for VC-IRSA.

Consistency measure (m1) (m2)

P

XiðyÞ Yes Yes

0P

XiðyÞ Yes Yes

lP

(11)

Theorem 5.4. Measures

P XP i ðyÞ and

P X6 i

ðyÞ have property (m4), i.e., for all P # C; XP i ;X 6 i #U; y 2 U:

8

y1;y22 U : y1DPy2)

PXP i ðy1Þ 6

P XP iðy2Þ;

8

y1;y22 U : y1DPy2)

P X6 iðy1Þ P

P X6 iðy2Þ: Unfortunately, measures

P XP i ðyÞ and

P X6 i

ðyÞ do not have property (m3). More precisely, for all P # C, XP i #X

P

j #U, j 6 i,

y 2 U, measure

P XP

iðyÞ is not monotonically non-increasing w.r.t. set of objects X

P

i , and for all P # C, X 6 i #X 6 j #U, j P i, y 2 U, measure

P X6 i

ðyÞ is not monotonically non-increasing w.r.t. set of objects X6

i. This can be illustrated by the following

example. We have P ¼ fa1g, X1¼ fy1g, X2¼ fy2g, X3¼ fy3g, where f ðy1;a1Þ ¼ 3, f ðy2;a1Þ ¼ 1, f ðy3;a1Þ ¼ 2. Moreover, let

us assume that attribute a1is gain-type and decision classes are ordered such that class X3is better than X2, which is better

than X1. We have,

P_XP 3

ðy3Þ ¼12<

PXP 2

ðy3Þ ¼ 1. The same can be shown for downward unions.

In order to ensure property (m3), in Sections5.2 and 5.3we introduce two possible modiﬁcations of measures

P XP i ðyÞ and

P X6 i ðyÞ. 5.2. Consistency measure

Cost-type consistency measures

P XP i ðyÞ and

P X6 i ðyÞ, for P # C, XP i ;X 6 i #U, y 2 U, are deﬁned as

P XP

i ðyÞ ¼ maxj6i

P XP

j ðyÞ; ð24Þ

P

X6

iðyÞ ¼ maxjPi

P X6 jðyÞ: ð25Þ Measure

P XP i ðyÞ (or

P X6 i

ðyÞ) is deﬁned as a maximal value of measure

P XP

i ðyÞ (

P

X6 i

ðyÞ) over all unions of decision classes which contain considered union XP

i (X 6 i). Measures

PXP i ðyÞ and

P X6 i

ðyÞ satisfy all monotonicity properties of

P XP i ðyÞ and

P X6 i ðyÞ, respectively. Moreover, as we show below, they have also monotonicity property (m3).

P XP

iðyÞ and

P X6

iðyÞ have property (m1), i.e., for all P # P

0_#_{C; X}P i ;X 6 i #U; y 2 U:

P XP i ðyÞ P

P0 XP i ðyÞ;

P X6 iðyÞ P

P XP

i ðyÞ has property (m2). More precisely, for all P # C, X

P i #U, X 0P i ¼ X P i [ X DP i , X DP i \ U ¼ ;, y 2 U:

P XP i ðyÞ ¼

P X6

iðyÞ has property (m2). More precisely, for all P # C, X

6 i #U, X 06 i ¼ X 6 i [ X D6 i , X D6 i \ U ¼ ;, y 2 U:

P X6 iðyÞ ¼

P X06 i ðyÞ: Theorem 5.8. Measure

P XP

i ðyÞ has property (m3), i.e., for all P # C, X

P i #X P j #U, j 6 i, y 2 U:

P XP i ðyÞ P

P XP j ðyÞ: Theorem 5.9. Measure

P X6 i

ðyÞ has property (m3), i.e., for all P # C, X6 i #X 6 j #U, j P i, y 2 U:

P X6 iðyÞ P

P X6 jðyÞ: Theorem 5.10. Measures

P XP i ðyÞ and

P X6 i

8

y1;y22 U : y1DPy2)

PXP i ðy1Þ 6

P XP iðy2Þ;

8

y1;y22 U : y1DPy2)

PX6 iðy1Þ P

P X6 iðy2Þ:

Monotonic P-lower approximation of union of classes XP i , X

6

i deﬁned according to(4)takes the form: Pb XP i ðXP i Þ ¼ fy 2 X P i :

PXP iðyÞ 6 b XP ig; ð26Þ Pb X6 iðX6 iÞ ¼ fy 2 X 6 i :

PX6 iðyÞ 6 b X6 ig; ð27Þ where cost-threshold b XP i;b X6

i 2 ½0; 1 reﬂects the highest degree of consistency acceptable to include object y in the P-lower approximation of union of classes XP

i , X 6

(12)

Theorem 5.11. Lower approximations deﬁned according to(26) and (27)satisfy condition(6): PðXP i Þ # P b XP i ðXP i Þ; PðX6 iÞ # P b X6 iðX6 iÞ: 5.3. Consistency measure

0

Another way to overcome the lack of property (m3) of

P XP

i ðyÞ and

P X6

i

ðyÞ is to consider cost-type consistency measures

0P XP i ðyÞ and

0P X6 i ðyÞ. For P # C, XP i , X 6 i, X 6 i1, X P

iþ1#U, y 2 U, they are deﬁned as

0P XP i ðyÞ ¼ jDþ PðyÞ \ X 6 i1j jXPi j ;

0P_X6 iðyÞ ¼ jDPðyÞ \ X P iþ1j jX6 ij : ð28Þ Consistency measure

0P XP i ðyÞ (or

0P X6 i

ðyÞ) is deﬁned as a ratio of the number of objects that belong both to dominance cone Dþ

PðyÞ (D

PðyÞ) and union X 6 i1(X

P

iþ1), to the number of objects belonging to union X P i (X

6

i). In other words, this measure

rep-resents the ratio of objects z 2 U that are counterexamples to the implication z 2 Dþ

PðyÞ (z 2 DPðyÞ) implies z 2 X P i (z 2 X

6 i) to

the total number of objects in XP i (X 6 i). Theorem 5.12. Measures

0P XP i ðyÞ and

0P X6 i

ðyÞ have property (m1), i.e., for all P # P0_#_{C; X}P i ;X 6 i #U; y 2 U:

0P XP i ðyÞ P

0P0 XP i ðyÞ;

0P X6 iðyÞ P

0P0 X6 iðyÞ: Theorem 5.13. Measure

0P XP

i ðyÞ has property (m2). More precisely, for all P # C, X

P i #U, X 0P i ¼ X P i [ X DP i , X DP i \ U ¼ ;, y 2 U:

0P XP i ðyÞ >

0P X0P i ðyÞ: Theorem 5.14. Measure

0P X6 i

ðyÞ has property (m2). More precisely, for all P # C, X6

i #U, X06i ¼ X 6 i [ X D6 i , X D6 i \ U ¼ ;, y 2 U:

0P X6 iðyÞ >

0P X06 i ðyÞ: Theorem 5.15. Measure

0P XP i

ðyÞ has property (m3), i.e., for all P # C, XP i #X P j #U, j 6 i, y 2 U:

0P XP i ðyÞ P

0P XP j ðyÞ: Theorem 5.16. Measure

0P X6 i

ðyÞ has property (m3), i.e., for all P # C, X6 i #X 6 j #U, j P i, y 2 U:

0P X6 iðyÞ P

0P X6 jðyÞ: Theorem 5.17. Measures

0P XP i ðyÞ and

0P X6 i

8

y1;y22 U : y1DPy2)

0PXP i ðy1Þ 6

0P XP i ðy2Þ;

8

y1;y22 U : y1DPy2)

0PX6 iðy1Þ P

0P X6 iðy2Þ: Measure

0P XP i ðyÞ (or

0P X6 i

ðyÞ) has different interpretation from consistency measures

P XP i ðyÞ (

P X6 i ðyÞ) and

P XP i ðyÞ (

P X6 i ðyÞ). It lacks likelihood explanation that is appropriate for the other two measures. It relates two rather antagonistic concepts. According to the deﬁnition of

0P

XP i ðyÞ (

0P X6

i

ðyÞ), the number of objects in the dominance cone of considered object y that do not belong to the considered union of classes is divided by the cardinality of the considered union of classes. This may result in low values of consistency measure

0P

XP i

ðyÞ (

0P X6

i

ðyÞ) for unions of classes XP i (X

6

i) that have high cardinality.

6

i deﬁned according to(4)takes the form: Pb 0 XP i ðXP i Þ ¼ fy 2 X P i :

0P XP i ðyÞ 6 b 0 XP i g; ð29Þ Pb 0 X6 iðX6 i Þ ¼ fy 2 X 6 i :

0P X6 iðyÞ 6 b 0 X6 ig; ð30Þ where cost-threshold b0 XP i 2 0; jX6 i1j jXP ij h i , b0 X6 i 2 0; jXP iþ1j jX6 ij

reﬂects the highest degree of consistency acceptable to include object y in the P-lower approximation of union of classes XP

i , X 6

(13)

Theorem 5.18. Lower approximations deﬁned according to(29) and (30)satisfy condition(6): PðXP i Þ # P b0 XP i ðXP i Þ; PðX6 iÞ # P b0 X6 iðX6 iÞ: 5.4. Consistency measure

l

For P # C, XP i ;X 6

i #U, y 2 U, we also consider the following gain-type consistency measures:

l

P XP i ðyÞ ¼ max R # P; z2D RðyÞ\XPi jDþ RðzÞ \ X P i j jDþ RðzÞj ; ð31Þ

l

P X6 i ðyÞ ¼ max R # P; z2Dþ RðyÞ\X 6 i jDRðzÞ \ X 6 ij jDRðzÞj : ð32Þ Measure

l

P XP i ðyÞ (or

l

P X6 i

ðyÞ) is deﬁned as a maximum rough membership to union XP i (X

6

i) over all subsets R of the set of

attributes P and over all objects z dominated by y (dominating y) and belonging to XP i (X

6

i). Comparing the above deﬁnitions

with the analogous deﬁnition(21)presented for VC-IRSA, one can easily observe that they have a new ingredient – the max-imum is calculated not only over all subsets R of P but also over all objects belonging to the intersection of the particular dominance cone of object y and the considered union of decision classes. Such a formulation ensures monotonicity property (m4), which is proved later in this section.

l

P XP i ðyÞ and

l

P X6 i

ðyÞ have property (m1), i.e., for all P # P0_#_{C; X}P i ;X 6 i #U; y 2 U:

l

P XP i ðyÞ 6

l

P0 XP iðyÞ;

l

P X6 iðyÞ 6

l

P XP i

ðyÞ has property (m2), i.e., for all P # C, XP

i #U, X0Pi ¼ X P i [ X DP i , X DP i \ U ¼ ;, y 2 U:

l

P XP i ðyÞ 6

l

P X6 i

ðyÞ has property (m2), i.e., for all P # C, X6 i #U, X 06 i ¼ X 6 i [ X D6 i , X D6 i \ U ¼ ;, y 2 U:

l

P X6 iðyÞ 6

l

P X06 i ðyÞ: Theorem 5.22. Measure

l

P XP

iðyÞ has property (m3), i.e., for all P # C, X

P i #X P j #U, j 6 i, y 2 U:

l

P XP i ðyÞ 6

l

P XP jðyÞ: Theorem 5.23. Measure

l

P X6 i

ðyÞ has property (m3), i.e, for all P # C, X6 i #X 6 j #U, j P i, y 2 U:

l

P X6 iðyÞ 6

l

P X6 jðyÞ: Theorem 5.24. Measures

l

P XP i ðyÞ and

l

P X6 i

8

y1;y22 U : y1DPy2)

l

P XP i ðy1Þ P

l

P XP i ðy2Þ;

8

y1;y22 U : y1DPy2)

l

PX6 iðy1Þ 6

l

P X6 iðy2Þ:

6

i deﬁned according to(2)takes the form: PaXP i ðXP i Þ ¼ fy 2 X P i :

l

P XP i ðyÞ P

a

X P i g; ð33Þ PaX6 iðX6 i Þ ¼ fy 2 X 6 i :

l

P X6 iðyÞ P

a

X 6 ig; ð34Þ where gain-threshold

a

XP i;

a

X 6

i 2 ½0; 1 reﬂects the lowest degree of consistency acceptable to include object y in the P-lower approximation of union of classes XP

i , X 6

(14)

Theorem 5.25. Lower approximations deﬁned according to(33) and (34)satisfy condition(5): PðXP i Þ # P a_XP i ðXP i Þ; PðX6 iÞ # P a_X6 i ðX6 iÞ:

In[2], we considered gain-type consistency measures

l

P XP

i

ðyÞ and

l

P X6

i

ðyÞ, which are deﬁned analogously to

l

P XP i ðyÞ and

l

P X6 i ðyÞ. For P # C, XP i ;X 6 i #U, y 2 U:

l

P XP i ðyÞ ¼ minRP; z2Dþ RðyÞ\X P i jDþRðzÞ \ X P i j jDþRðzÞj ;

l

P_X6 iðyÞ ¼ minRP; z2D RðyÞ\X 6 i jDRðzÞ \ X 6 ij jDRðzÞj :

It appears that these measures have properties (m1) and (m4) while they do not have properties (m2) and (m3). Therefore, we refrained from using them in the deﬁnition of the P-lower approximation.

5.5. Summary

In this section, we introduced several consistency measures for VC-DRSA. Their properties are summarized inTable 2. Remark that

P XP i ðyÞ and

P X6 i

ðyÞ are the only measures missing desirable property (m3). Therefore, two possible modiﬁcations of these measures, denoted by

P

XP i ðyÞ,

P X6 i ðyÞ and

0P XP i ðyÞ,

0P X6 i

ðyÞ, were further investigated. 6. Illustrative example

Let us consider VC-IRSA and the set of objects shown inFig. 3.

First, let us determine the P-lower approximation of class X2 using deﬁnition(18), for bX2¼ 0. We can observe that P0_ðX

2Þ ¼ fy2;y3g. Object y1is not included in P0ðX2Þ because

PX2ðy1Þ ¼

1

3. We can also notice that

fa1g X2 ðy2Þ ¼

fa2g X2 ðy2Þ ¼ 1 3, while

P

X2ðy2Þ ¼ 0. This illustrates property (m1) of measure

P

XiðyÞ. In order to exemplify property (m2) of measure

P

XiðyÞ, we extend class X2by adding to this class new object y7, with the following description: a1¼ 1, a2¼ 1. One can easily verify that values

of measure

P

XiðyÞ for class X2and objects y1, y2and y3do not change.

Second, let us calculate the P-lower approximation of class X2 using deﬁnition(20), for b0X2¼ 0. We can notice that P0_ðX

2Þ ¼ fy2;y3g. Object y1is not included in P0ðX2Þ because

0PX2ðy1Þ ¼

1

3. We can also notice that

0fa1g X2 ðy2Þ ¼

0fa2g X2 ðy2Þ ¼ 1 3, Table 2

Monotonicity of consistency measures deﬁned for VC-DRSA.

Consistency measure (m1) (m2) (m3) (m4) P XP i ðyÞ, P X6 i

ðyÞ Yes Yes No Yes

P XP i ðyÞ, P X6 i

ðyÞ Yes Yes Yes Yes

0P XP i ðyÞ, 0P X6 iðyÞ

Yes Yes Yes Yes

lP XP

iðyÞ,l

P X6

iðyÞ Yes Yes Yes Yes

a1 a2 ● ● ● ● ● ● ● ● ● ● ● ●

●

0 1 2 0 1 2 y1 y2 y3 y4 y5 y6 X2 X2 X2 X1 X1 X1

(15)

while

0P

X2ðy2Þ ¼ 0. This illustrates property (m1) of measure

0P

XiðyÞ. In order to exemplify property (m2) of measure

0P XiðyÞ, we extend class X2by adding to this class new object y7, with the following description: a1¼ 1, a2¼ 1. One can easily verify that

values of measure

0P

XiðyÞ for class X2and objects y1, y2and y3decrease. Third, let us consider measure

l

P

XiðyÞ and deﬁnition(22). For

a

X2¼

2

3we have P

2

3ðX₂Þ ¼ fy

1;y2;y3g. It is worth noting that

because

l

fa1g X2 ðy1Þ ¼ 2 3,

l

fa2g X2 ðy1Þ ¼ 1 2and

l

P X2ðy1Þ ¼ 1 2, then

l

P X2ðy1Þ ¼ 2

3. Thus, we can observe that measure

l

P

XiðyÞ has property (m1). Let us now extend the set of objects fromFig. 3with object y72 X2, for which a1¼ 1, a2¼ 1, as we did in the ﬁrst

exam-ple. Thus, we obtain

l

P X2ðy1Þ ¼

3

4P23. This shows that measure

l

PXiðyÞ has property (m2).

Now, let us apply VC-DRSA to the set of objects presented inFig. 4. Class X3is better than class X2, which is better than

class X1. Thus, we may distinguish two upward unions of decision classes: XP3, X P

2, and two downward unions of decision

classes: X6 1, X

6 2.

First, let us determine the P-lower approximation of union XP

3 using deﬁnition(26), for bXP 3 ¼

1

3. We can observe that

P1 3ðXP

3Þ ¼ fy1;y2g. Object y3is not included in P

1 3ðXP 3Þ because

PXP 3ðy3Þ ¼ maxf 2 4; 0 1g ¼ 1 2> 1

3. We can notice that

fa1g XP 3 ðy2Þ ¼ 1 4,

fa2g XP 3 ðy2Þ ¼12, while

PXP 3ðy2Þ ¼ 1

4. This illustrates property (m1) of measure

PXP i

ðyÞ. In order to exemplify property (m2) of mea-sure

P

XP i

ðyÞ, we consider possible extension of union XP

3 by new object y82 X3, with the following description: a1¼ 5, a2¼ 6.

One can easily verify that in such a case values of measure

P XP

iðyÞ for extended union X

P

3 and objects y1, y2, y3do not change.

Let us also observe that

P XP

3

ðy3Þ ¼12, while

PXP 2

ðy3Þ ¼ 0. This illustrates property (m3) of measure

PXP i ðyÞ. Moreover,

P XP 3 ðy2Þ ¼14,

PXP 3

ðy3Þ ¼12and y2DPy3for P ¼ fa1;a2g, which shows that measure

P_XP i

ðyÞ has property (m4). Second, let us calculate the P-lower approximation of union XP

3 using deﬁnition (29), for b0XP 3 ¼

1

3. We have

P1 3ðXP

3Þ ¼ fy1;y2g. Object y3 is not included in P

1 3ðXP 3Þ because

0PXP 3ðy3Þ ¼ 2 3> 1

3. We can observe that

0fa1g XP 3 ðy2Þ ¼ 1 3,

0fa2g XP 3 ðy2Þ ¼ 2 3, while

0PXP 3ðy2Þ ¼ 1

3. This illustrates property (m1) of measure

0PXP

i ðyÞ. In order to exemplify property (m2) of mea-sure

0P

XP i

ðyÞ, we consider possible extension of union XP

3 by new object y82 X3, having the following description: a1¼ 5,

a2¼ 6. One can easily verify that in such a case values of measure

0PXP

iðyÞ for extended union X

P

3 and objects y1, y2, y3

de-crease, since jXP

3j increases. Let us also notice that

0PXP 3ðy3Þ ¼

2

3, while

0PXP

2ðy3Þ ¼ 0. This illustrates property (m3) of measure

0P XP i ðyÞ. Moreover,

0P XP 3 ðy2Þ ¼13,

0PXP 3

ðy3Þ ¼23and y2DPy3for P ¼ fa1;a2g, which shows that measure

0P_XP i

ðyÞ has property (m4). Third, let us consider measure

l

P

XP i

ðyÞ and corresponding deﬁnition(33). For

a

XP 3 ¼ 2 3we have P 2 3ðXP 3Þ ¼ fy1;y2g. It is worth

noting that because

l

fa1g XP 3 ðy1Þ ¼

l

faXP1g 3 ðy2Þ ¼23,

l

fa2g XP 3 ðy1Þ ¼

l

faXP2g 3 ðy3Þ ¼35and

l

PXP 3 ðy3Þ ¼35, then

l

PXP 3

ðy1Þ ¼23. Thus, we can observe

that measure

l

P XP

i ðyÞ has property (m1). In order to illustrate property (m2) of measure

l

P XP

iðyÞ, we consider possible exten-sion of union XP

3 by new object y82 X3, with the following description: a1¼ 5, a2¼ 6. Then, we have

l

P_XP 3ðy1Þ ¼

3 4>

2 3. Let us

also observe that

l

P XP 3ðy3Þ ¼ 3 5, while

l

P XP

2ðy3Þ ¼ 1. This exempliﬁes property (m3) of measure

l

P XP iðyÞ. Moreover,

l

P XP 3ðy2Þ ¼ 2 3>

l

P XP 3ðy3Þ ¼ 3

5, while y2DPy3for P ¼ fa1;a2g, which shows that measure

l

P_XP

i ðyÞ has property (m4). 7. Final remarks and conclusions

In this paper, we have presented several deﬁnitions of monotonic Variable Consistency Rough Set Approaches that employ indiscernibility or dominance relation. We have stressed the importance of some monotonicity properties of the consistency measure used in the deﬁnition of a lower approximation. We have considered the following monotonicity

0 1 2 3 4 5 6 7 01234567 a1 a2 ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●

y1 y2 y3 y4 y5 y6 y7 X3 X3 X3 X2 X2 X2 X1

(16)

properties: (m1) – monotonicity w.r.t. set of attributes, (m2) and (m3) – monotonicity w.r.t. set of objects (where (m2) cor-responds to growing universe U and (m3) to ﬁxed universe U with growing unions of decision classes), and (m4) – mono-tonicity w.r.t. dominance relation (for approaches based on dominance relation only).

We have proposed two types of measures enjoying the above monotonicity properties. The ﬁrst type stems from consis-tency measure

, which is a catch-all likelihood measure. This consistency measure has a comprehensible probabilistic explanation. It has also a close relation with the Bayes factor and conﬁrmation measure l. We proposed a kind of comple-mentary measure to

denoted by

0_{. One can observe that for}

_{, there is a tendency of including relatively more objects}

to lower approximations when the approximated class or union of classes has low cardinality. On the other hand, one can observe that for

0_{, there is a tendency of including relatively more objects to lower approximations when the}

approx-imated class or union of classes has high cardinality. Both of these measures are directly applicable in VC-IRSA. Unfortu-nately, in the context of VC-DRSA, measure

does not have property (m3). In order to overcome this problem, we introduced measure

_{that involves a speciﬁc scheme of computation of consistency measure}

_{over supersets of the}

con-sidered union of classes.

Monotonic measures of the second type stem from consistency measure

l

. They require to take into account all subsets of the set of considered attributes. Computation of lower approximations deﬁned by means of monotonic measure

l

is an NP-hard problem, equivalent to induction of a set of all rules. On the other hand, computation of such approximations and rule induction can be combined, and thus the total time would be of the same order as the time for induction of all rules.

As a conclusion, we can recommend using consistency measure

or

0_{for VC-IRSA and consistency measure}

_or

0_for

VC-DRSA. These measures have all required monotonicity properties and are much less computationally intensive than the monotonic measures of the second type.

Acknowledgement

The ﬁrst, third and fourth author wish to acknowledge ﬁnancial support from the Ministry of Science and Higher Educa-tion, grant N N519 314435.

Appendix.

Proof of Theorem 4.1. From the deﬁnition of rough granules IPðyÞ and IP0ðyÞ, P # P0#C, y 2 U,

IPðyÞ IP0ðyÞ

for Xi;:Xi#U being both independent of sets of considered attributes P and P0. This implies: jIPðyÞ \ :Xij j:Xij PjIP0ðyÞ \ :Xij j:Xij ()

P XiðyÞ P

P0 XiðyÞ:

Proof of Theorem 4.2. Since new objects are introduced to class Xi#U, thus for all sets of objects Xi#U, X0i¼ Xi[ XDi, where

XD

i \ U ¼ ;, :Xi¼ :X0i:

For all P # C, y 2 U, this implies:

jIPðyÞ \ :Xij j:Xij ¼jI 0 PðyÞ \ :X 0 ij j:X0 ij ()

P XiðyÞ ¼

P X0 iðyÞ;

where I0PðyÞ denotes a set of objects indiscernible with object y when considering set of attributes P and universe U [ X

D

i. h

Proof of Theorem 4.3. For each object y 2 Xi, IPðyÞ # Xiiff

PXiðyÞ ¼ 0. h Proof of Theorem 4.4. Analogous to Proof of Theorem4.1for measure

P

XiðyÞ – only the common denominators in fractions are changed from j:Xij to jXij. h

Proof of Theorem 4.5. New objects are introduced to class Xi#U. Thus, for all sets of objects Xi#U, X0i¼ Xi[ X

D

i, where

XD

i \ U ¼ ;,

:Xi¼ :X0i; jXij < jX0ij:

This implies that for all P # C, y 2 U:

jIPðyÞ \ :Xij jXij >jI 0 PðyÞ \ :X 0 ij jX0ij ()

0P XiðyÞ >

0P X0 iðyÞ; where I0

PðyÞ denotes a set of objects indiscernible with object y when considering set of attributes P and universe U [ X

D

(17)

0PXiðyÞ ¼ 0. h Proof of Theorem 4.7. For all P # P0_#_{C; X}

i#U; y 2 U,

l

P XiðyÞ ¼ max_{R # P} IRðyÞ \ Xi j j IRðyÞ j j 6maxR # P0 IRðyÞ \ Xi j j IRðyÞ j j ¼

l

P0 XiðyÞ:

Proof of Theorem 4.8. Let us consider P # C, Xi#U, X0i¼ Xi[ XDi, X

D

i \ U ¼ ;, y 2 U. Since all new objects are added to class

Xi, both numerator and denominator of fraction IPðyÞ \ Xi j j IPðyÞ j j ¼

l

P XiðyÞ

can increase only with the same number k P 0, equal to difference jI0

PðyÞj jIPðyÞj: IPðyÞ \ Xi j j þ k IPðyÞ j j þ k ¼ I0 PðyÞ \ X 0 i I0 PðyÞ ¼

l

PX0 iðyÞ; where I0

PðyÞ denotes a set of objects indiscernible with object y when considering set of attributes P and universe U [ X

D

i.

Further, let us introduce the following notation: a ¼ jIPðyÞ \ Xij, b ¼ jIPðyÞj, and let us notice that a 6 b. We can observe

that

l

P XiðyÞ 6

l

P X0 iðyÞ; ð35Þ

which is proved in the following way:

a b6 a þ k b þ k () aðb þ kÞ 6 bða þ kÞ () ab þ ak 6 ab þ bk () ak 6 bk () a 6 b: Thus,

l

P XiðyÞ ¼ max_{R # P}

l

R XiðyÞ 6 max_{R # P}

l

R X0 iðyÞ ¼

l

P X0 iðyÞ:

l

PXiðyÞ ¼ 1. h Proof of Theorem 5.1. From the deﬁnition of dominance cones Dþ

PðyÞ and D þ P0ðyÞ, P # P0#C, y 2 U, DþPðyÞ D þ P0ðyÞ for XP i ;X 6

i1#U being both independent of sets of considered attributes P and P

0_{. This implies:} jDþ PðyÞ \ X 6 i1j jX6 i1j PjD þ P0ðyÞ \ X 6 i1j jX6 i1j ()

P XP i ðyÞ P

P0 XP i ðyÞ:

The proof for downward union X6

i is analogical, but starts from the observation that for negative dominance cones DPðyÞ and

D P0ðyÞ, P # P0#C, y 2 U, D PðyÞ D P0ðyÞ:

Proof of Theorem 5.2. New objects are introduced to union of classes XP

i #U. Thus, for all sets of objects X P i #U, X0P i ¼ X P i [ X DP i , where X DP i \ U ¼ ;, X6 i1¼ X 06 i1:

For all P # C, y 2 U, this implies:

jDþPðyÞ \ X 6 i1j jX6 i1j ¼jD 0þ P ðyÞ \ X 06 i1j jX06 i1j ()

P XP i ðyÞ ¼

P X0P i ðyÞ; where D0þ

PðyÞ denotes P-positive dominance cone of object y when considering universe U [ X

DP i . h

Proof of Theorem 5.3. Analogous to Proof of Theorem5.2which is carried out for sets of objects XP

i and X0Pi . In this case, sets

of objects XP iþ1and X

0P

iþ1are considered instead of sets X 6 i1and X

06